Turn 2D Mesh into a 3D Mesh reuleaux solids
The wikipedia article says


  Four balls intersect to form a Reuleaux tetrahedron.


Which is easy to implement with boolean operations (and balls).


Start with a pyramid. To create it, activate the Extra Objects Addon.
  
In the 3D View, add (&#8679; ShiftA) a Geodesic Dome.
  
Add a subdivided cube cast into a sphere and snap it to the vertices of the pyramid.

Select one of the cubes and add three boolean modifiers. Choose the other three cubes as inputs.

The distance of the centers of the spheres is the length of an edge of the pyramid. The radius of the spheres should equal this value. (The dimensions (diameters in this case) will naturally be twice the value.) In my case the pyramid has an edge length of 1.633, hence the spheres have to be rescaled to match a dimension of 1.633 * 2.

Hide all the objects, except the  one with the boolean modifiers. Add an edge split modifier to clean up the seams. I used a high subdivision value (4).